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safety proof of EWD687a #215

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6 changes: 6 additions & 0 deletions specifications/SpecifyingSystems/AdvancedExamples/Graphs.tla
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Original file line number Diff line number Diff line change
@@ -1,4 +1,10 @@
------------------------------- MODULE Graphs -------------------------------
(***************************************************************************)
(* Representation of (directed) graphs in TLA+. *)
(* This is the ancestor of the Graphs community module, available at *)
(* https://github.com/tlaplus/CommunityModules/blob/master/modules/Graphs.tla *)
(* which contains more operators and also slightly changed definitions. *)
(***************************************************************************)
LOCAL INSTANCE Naturals
LOCAL INSTANCE Sequences

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46 changes: 23 additions & 23 deletions specifications/ewd687a/EWD687a.tla
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Expand Up @@ -87,7 +87,7 @@
(* not to be detected. It requires that a process can send acknowledgment *)
(* messages to any process that can send it computation messages. *)
(***************************************************************************)
EXTENDS Integers
EXTENDS Integers, Graphs

(***************************************************************************)
(* We declare the following constants: *)
Expand Down Expand Up @@ -215,7 +215,7 @@ VARIABLES
(* network. In an implementation, we would, thus, expect to find *)
(* code that maintains the values of active, sentUnacked, and *)
(* rcvdUnacked in the implementation of a process. The variables msgs *)
(* and acks, however, would likely be implicitly because they model *)
(* and acks, however, would likely be implicit because they model *)
(* the state of the network. *)
(***********************************************************************)

Expand All @@ -231,7 +231,7 @@ VARIABLES
msgs,
(***********************************************************************)
(* For an edge <<q,p>> \in OutEdges(p), acks[<<q,p>>] is the number of *)
(* control) messages in transit from process p to process q. *)
(* (control) messages in transit from process p to process q. *)
(***********************************************************************)
acks
(***********************************************************************)
Expand Down Expand Up @@ -330,7 +330,7 @@ RcvAck(p) == \E e \in OutEdges(p) :
(* 2b. The receiver q of the ack is the parent of p in the overlay tree, *)
(* and p is neutral after sending the ack. *)
(* *)
(* UP. If process p is neutral after sending an ack (hence netural(p)'), *)
(* UP. If process p is neutral after sending an ack (hence neutral(p)'), *)
(* it removes itself from the overlay tree by setting the value of *)
(* upEdge[p] to NotAnEdge. *)
(***************************************************************************)
Expand Down Expand Up @@ -383,16 +383,17 @@ RcvMsg(p) == \E e \in InEdges(p) :
(* that p was not a node of the overlay tree when it became idle. Thus, *)
(* there is no need to change upEdge[p] in the Idle subaction. *)
(***************************************************************************)
Idle(p) == /\ active' = [active EXCEPT ![p] = FALSE]
Idle(p) == /\ active[p] \* otherwise this step is subsumed by stuttering
/\ active' = [active EXCEPT ![p] = FALSE]
/\ UNCHANGED <<msgs, acks, sentUnacked, rcvdUnacked, upEdge>>

----------------------------------------------------------------------------

Next == \E p \in Procs : SendMsg(p) \/ SendAck(p) \/ RcvMsg(p) \/ RcvAck(p)
\/ Idle(p)

Spec == Init /\ [][Next]_vars /\ WF_vars(Next)

----------------------------------------------------------------------------

(***************************************************************************)
Expand All @@ -401,34 +402,33 @@ Spec == Init /\ [][Next]_vars /\ WF_vars(Next)
(* rcvdUnacked[e] equals sentUnacked[e], for any e in Edges. *)
(***************************************************************************)
CountersConsistent ==
[] \A e \in Edges: sentUnacked[e] = rcvdUnacked[e] + acks[e] + msgs[e]
\A e \in Edges: sentUnacked[e] = rcvdUnacked[e] + acks[e] + msgs[e]

THEOREM Spec => CountersConsistent
THEOREM Spec => []CountersConsistent

(***************************************************************************)
(* The overlay tree is a tree of non-neutral nodes rooted in the leader, *)
(* or the tree is empty s.t. it has no edges nor nodes. *)
(* With bigger graphs, expect a significant slowdown when model-checking *)
(* the property TreeWithRoot. *)
(* The overlay tree is a tree rooted in the leader such that all inner *)
(* nodes are non-neutral. *)
(* *)
(* In a variant of this spec, the conjunct labelled Up in the SendAck *)
(* subactions could be removed. In this variant, the processes in the *)
(* overlay tree would be processes that are or were non-neutral in the *)
(* past. Consequently, the conjunct labelled N would have to be *)
(* removed from the property TreeWithRoot . *)
(* removed from the property TreeWithRoot. *)
(***************************************************************************)

TreeWithRoot ==
LET E == {upEdge[p] : p \in DOMAIN upEdge} \ {NotAnEdge}
N == {e[1] : e \in E} \cup {e[2] : e \in E}
G == INSTANCE Graphs
O == G!Transpose([edge |-> E, node |-> N])
IN [](
\* O is a tree rooted in the leader.
/\ O.edge # {} => G!IsTreeWithRoot(O, Leader)
\* All nodes of the overlay tree are non-neutral.
/\ N:: \A n \in O.node: ~neutral(n))
\* The original definition includes every node occurring in the tree
\* twice (except the leader), which unnecessarily complicates the proof.
N == (* {e[1] : e \in E} \cup *) {e[2] : e \in E} \cup {Leader}
O == Transpose([edge |-> E, node |-> N])
IN \* O is a tree rooted in the leader.
/\ IsTreeWithRoot(O, Leader)
\* All nodes of the overlay tree, except the root, are non-neutral.
/\ N:: \A n \in O.node \ {Leader} : ~neutral(n)

THEOREM Spec => TreeWithRoot
THEOREM Spec => []TreeWithRoot

---------------------------------------------------------------------------

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